Matematica e Fisica al crocevia

Transcription

1 Matematica e Fisica al crocevia Roberto Longo Colloquium di Macroarea, parte I Roma, 9 Maggio 2017

2 The cycle Phys-Math-Math-Phys Nessuna humana investigazione si può dimandara vera scienzia s essa non passa per le matematiche dimonstrazioni. Leonardo da Vinci, Trattato della Pittura, 1500 circa Physics Mathematics experiment theory art formalism Physics Mathematics The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Eugene Wigner, Comm. Pure Appl. Math. 1960

3 The unity of Mathematics Euler s formula: e: Real Analisys i: Complex Analysis π: Geometry 1: Algebra e iπ + 1 = 0 Cartesio: Geometry to Algebra: x 2 + y 2 = r 2 Now x, y, r can be generalised!

4 The unity of Physics Einstein: E = mc 2 Hawking effect: a quantum black hole has a surface temperature c: speed of light G: gravitational constant k: Boltzmann constant : Planck constant T = c3 8πkMG Grand Unification and the dream of Quantum Gravity, see part II by Massimo Bianchi

5 From classical to quantum physics Until the XIX century, Mathematics and Physics had a natural and continuous interplay: Archimede, Galileo, Newton, Gauss, etc. At the beginning of the past century, Quantum Physics makes a discontinuity and opens a completely new frame.

6 Quantum Mechanics Planck Schroedinger Heisenberg von Neumann Schrödinger: i ψ(x, t) = Hψ(x, t) t Differential equations Heisenberg: PQ QP = i I Linear operators on a Hilbert space, noncommutativity is essential!

7 von Neumann: The two QM are equivalent, uniqueness of CCR Operator Algebras: Classical Commutative Manifold X C (X ) Topological space X C(X ) Measure space X L (X, µ) Quantum Noncommutative -algebra A C -algebra A von Neumann algebra A

8 Thermal equilibrium states Thermodynamics concerns heat and temperature and their relation to energy and work. A primary role is played by the equilibrium distribution. Gibbs states Finite quantum system: A matrix algebra with Hamiltonian H and evolution τ t = Ade ith. Equilibrium state ϕ at inverse temperature β is given by the Gibbs property ϕ(x ) = Tr(e βh X ) Tr(e βh ) What are the equilibrium states at infinite volume where there is no trace, no inner Hamiltonian?

9 von Neumann algebras H a Hilbert space. B(H) algebra of all bounded linear operators on H. M B(H) is a von Neumann algebra if it is a -algebra and is weakly closed. Equivalently (von Neumann density theorem) M = M with M = {T B(H) : TA = AT A M} the commutant. A C -algebra is only closed in norm. Observables are elements A of M, states are normalised positive linear functionals ϕ, ϕ(a) = expected value of the observable in the state M abelian M = L (X, µ).

10 KMS states (HHW, Baton Rouge conference 1967) Infinite volume. A a C -algebra, τ a one-par. automorphism group of A. A state ϕ of A is KMS at inverse temperature β > 0 if for X, Y A ϕ ( X τ t+iβ (Y ) ) = ϕ ( τ t (Y )X ) ϕ ( τ t (Y )X ) β ϕ ( X τ t (Y ) ) KMS = thermodynamical equilibrium condition

11 Modular theory and Connes cocycles Let M be a von Neumann algebra and ϕ a normal faithful state on M. The Tomita-Takesaki theorem gives a canonical evolution: t R σ ϕ t Aut(M) Non commutative measure theory is dynamical! By a remarkable historical coincidence, Tomita announced the theorem at the 1967 Baton Rouge conference. Soon later Takesaki charcterised the modular group by the KMS condition. The Connes Radon-Nikodym cocycle relates the modular groups of different states u t = (Dψ : Dϕ) t M, σ ψ t = u t σ ϕ t ( )u t

12 Jones index Factors (von Neumann algebras with trivial center) are very infinite-dimensional objects. For an inclusion of factors N M the Jones index [M : N ] measure the relative size of N in M. Surprisingly, the index values are quantised: [M : N ] = 4 cos 2( π ), n = 3, 4,... or [M : N ] 4 n Jones index appears in many places in math and in physics. 2cos(π/10)

13 Quantum Field Theory In QFT we have a quantum system with infinitely many degrees of freedom. The system is relativistic and there is particle creation and annihilation. No mathematically rigorous QFT model with interaction still exists in 3+1 dimensions! Haag local QFT: O spacetime regions von Neumann algebras A(O) to each region one associates the noncommuative functions with support in O.

14 Local conformal nets A local net A on the circle S 1 is a map interval I von Neumann algebra A(I ) Isotony. I 1 I 2 = A(I 1 ) A(I 2 ) Locality. I 1 I 2 = = [A(I 1 ), A(I 2 )] = {0} Diffeomorphism covariance with positive energy and vacuum vector. I è A(I) Noncommutative chart

15 Representations A (DHR) representation ρ of local conformal net A on a Hilbert space H is a map I I ρ I, with ρ I a normal rep. of A(I ) on H s.t. ρĩ A(I ) = ρ I, I Ĩ, I, Ĩ I. Index-statistics theorem (R.L.): d(ρ) = [ ( ρ I A(I ) ) ( ) ] 1 2 : ρi A(I ) DHR dimension = Jones index Physical index Anal. index

16 Classification of local conformal nets, c = 1 6 m(m+1) Local conformal nets with c < 1 are classified by pair of Dynkin diagrams A D 2n E 6,8 with difference of Coxeter numbers 1 (Kawahigashi, L. 2004) m Labels for Z n (A n 1, A n ) 4n + 1 (A 4n, D 2n+2 ) 4n + 2 (D 2n+2, A 4n+2 ) 11 (A 10, E 6 ) 12 (E 6, A 12 ) 29 (A 28, E 8 ) 30 (E 8, A 30 ) Four exceptional cases, one new example (A 28, E 8 ), probably not constructable as coset Case c = 1 classified by Xu, Carpi (with a spectral condition) Many new models by mirror symmetry, F. Xu.

17 Towards a QFT index theorem D elliptic differential operator between vector bundles E and F over a compact manifold X. Atiyah-Singer index theorem: Analytical index(d) = Topological index(d) The Index Theorem is one of the most influential theorem in Mathematics and in Physics of the past century. Is there an index theorem with infinitely many degrees of freedom?

18 From classic to QFT CLASSICAL QUANTUM Classical variables Differential forms Chern classes Quantum geometry Fredholm operators Index Cyclic cohomology Variational calculus Infinite dimensional manifolds Functions spaces Wiener measure Subfactors Bimodules, Endomorphisms Jones index Supersymmetric QFT

19 An example: topological sectors (F. Xu, R.L.) A local conformal net, f : S 1 S 1, deg(f ) = n, B (A A) Zn τ f topological sector of B associated with f. We have: Jones index(τ f ) = µ deg(f ) A Anal. index Topol. index Physical sector

20 Second example: incremental free energy in QFT A QFT index theorem holds for a certain class of black holes (e.g. Rindler or Schwarschild) (F ρ1 F ρ2 ) = β 1( log d(ρ 1 ) log d(ρ 2 ) ) (F ρ1 F ρ2 ): incremental free energy adding the charge ρ 1 and removing the charge ρ 2 β 1 : Hawking-Unruh temperature (Bisognano-Wichmann KMS modular property for a uniformly accelerated observer) d(ρ 1 ), d(ρ 2 ): DHR statistical dimensions: integers! (Analog to the Fredholm index)

21 Thanks, now Part II